M03_COPI1396_13_SE_C03.QXD 10/12/07 11:54 AM Page 96 96 CHAPTER 3 Language and Definitions one’s person—satisfy the meaning of “carry” in that law? Justice Stephen Breyer argued that Congress intended the word in its ordinary, everyday meaning, without the artificial limitation that it be immediately accessible. Quoting Robinson Crusoe and Moby Dick, he pointed to the common use of “carry” to mean “convey in a vehicle.” The mandatory sentence, he con- cluded, is thus properly imposed. Justice Ruth Bader Ginsburg found Breyer’s literary evidence selective and unpersuasive; in response, she offered quotations from Rudyard Kipling, the TV series M.A.S.H., and President Theodore Roosevelt’s “Speak softly and carry a big stick” to show that “carry” is properly understood in the federal statute to mean “the gun at hand, ready for use as a weapon” [Muscarello v. U.S., U.S. 96-1654 (1998)]. In this controversy, which side puts forward the better precising definition? 3.5 The Structure of Definitions: Extension and Intension A definition states the meaning of a term. When we look closely at the literal (or descriptive) meaning of a term, however, we see that there are different senses in which that term has meaning. With those different senses distinguished (our object just below), we will also see that definitions may be grouped and under- stood not only on the basis of their functions (as in the preceding section), but in view of the way those definitions are built: their structure. We focus on general terms—terms that are applicable to more than one object—which are of critical importance in reasoning. The word “planet” is a typical general term; it is applicable to a number of objects, and it applies in the same sense equally to Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune.* What is meant by the word “planet” is (in one sense) that set of objects. The collection of planets constitutes the meaning of the term, its extensional meaning. If I say that all planets have elliptical orbits, part of what I assert is that Mars has an elliptical orbit, and another part is that Venus has an elliptical orbit, and so on. The extension of the general term “planet” consists of the objects to which the term may be correctly applied. The extensional meaning (also called the denotative meaning) of a general term is the collection of the ob- jects that constitutes the extension (or denotation) of the term. To understand the meaning of a general term is to know how to apply it correctly. But it is not necessary to know all of the objects to which it may be applied correctly in order to apply it correctly. All the objects within the exten- sion of a given term have some common attributes or characteristics that lead us *But not to Pluto! As explained in the preceding section, Pluto is now classified by the International Astronomical Union as a “dwarf planet.” M03_COPI1396_13_SE_C03.QXD 10/12/07 11:54 AM Page 97 3.5 The Structure of Definitions: Extension and Intension 97 to use the same term to denote them. If we know these attributes, we may know the meaning of a term in a different sense, without knowing its exten- sion. In this second sense, meaning supposes some criterion for deciding, with respect to any given object, whether it falls within the extension of that term. This sense of meaning is called the intensional meaning (or, sometimes, connotative meaning) of the term. The set of attributes shared by all and only those objects to which a general term refers is called the intension (or connotation) of that term. Every general term has both an intensional (or connotative) meaning and an extensional (or denotative) meaning. Consider the general term “skyscraper.” It applies correctly to all buildings over a certain height; that is its intension. The ex- tension of the term “skyscraper” is the class of buildings that contains the Empire State Building in New York, the Sears Tower in Chicago, the Shanghai World Financial Center, the Petronas Twin Towers in Kuala Lumpur, and oth- ers also—that is, the collection of the objects to which the term applies. The extension of a term (its membership) is determined by its intension. The intension of the term “equilateral triangle” is the attribute of being a plane figure enclosed by three straight lines of equal length. The extension of “equi- lateral triangle” is the class of all those objects, and only those objects, that have this attribute. Because any object that has this attribute must be a mem- ber of that class, we say that the term’s intension determines its extension. However, the reverse is not true: The extension of a term does not deter- mine its intension. Consider “equiangular triangle,” which has an intension different from that of “equilateral triangle.” The intension of “equiangular triangle” is the attribute of being a plane figure enclosed by three straight lines that intersect each other to form equal angles. It is true, of course, that the extension of the term “equiangular triangle” is exactly the same as the exten- sion of the term “equilateral triangle.” So if we were to identify the extension of one of these terms, that would leave the intension of the class uncertain; intension is not determined by extension. Terms may have different intensions and the same extension; but terms with different extensions cannot possibly have the same intension. When attributes are added to the intension of a term, we say that the in- tension increases. Begin with a general term such as “person.” Add “living.” Add “over twenty years old.” Add “born in Mexico.” With each such addition the intension increases; the intension of the term “Living person over twenty years old born in Mexico” is far greater than that of “person.” So these terms are given here in order of increasing intension. However, increasing their in- tention decreases their extension. The number of living persons is much lower than that of persons, and the number of living persons over twenty years old is lower still, and so on. M03_COPI1396_13_SE_C03.QXD 10/12/07 11:54 AM Page 98 98 CHAPTER 3 Language and Definitions One may be tempted to say that extension and intension always vary in- versely, but in fact that is not the case. This is because there comes a point when increasing the intension of the term has no effect on its extension. Consider this series: “living person,” “living person with a spinal column,” “living person with a spinal column less than one thousand years old,” “living person with a spinal column less than one thousand years old who has not read all the books in the Library of Congress.” These terms are clearly in order of increasing intension, but the extension of each of them is exactly the same, not decreasing at all. So we can say that, if terms are arranged in order of in- creasing intension, their extensions will be in nonincreasing order. That is, if ex- tensions vary, they will vary inversely with the intensions. Note that the extensions of some terms are empty; there simply are no ob- jects having the indicated attributes. In Greek mythology, Bellerophon killed the fire-breathing Chimera, a monster with a lion’s head, a goat’s body, and a serpent’s tail. We fully understand the intension of the term Chimera, but it has no extension. Some bad arguments play on the fact that meaning can refer to extension or to intension, while extension may be empty. For example: The word “God” is not meaningless; therefore it has a meaning. But by defini- tion, the word “God” means a being who is all-powerful and supremely good. Therefore that all-powerful and supremely good being, God, must exist. The word “God” is certainly not meaningless, and so there is an intension that is its meaning. But it does not follow from the fact that a term has an intension that it denotes any existent thing.* A contemporary critic has argued in similar fashion: Kitsch is the sign of vulgarity, sleaze, schlock, sentimentality, and bad faith that mark and mar our human condition. That is why utopia can be defined as a state of affairs in which the term has disappeared because it no longer has a referent.14 Here the writer has failed to distinguish between meaning and referent. Many valuable terms—those naming mythological creatures, for example—have no existing referent, no extension, but we do not want or expect such terms to dis- appear. Terms with intension but no extension are very useful. If utopia some- day comes, we may wish to express our good fortune in having eliminated “kitsch” and “sleaze,” but to do that we will need to be able to use those very words meaningfully. *The useful distinction between intension and extension was introduced and emphasized by St. Anselm of Canterbury (1033–1109), who is best known for his “ontological argument”— to which the preceding fallacious argument has little resemblance. M03_COPI1396_13_SE_C03.QXD 10/12/07 11:54 AM Page 99 3.5 The Structure of Definitions: Extension and Intension 99 We now use the distinction between intension and extension to explain some techniques for constructing definitions. Some definitions approach a gen- eral term by focusing on the class of objects to which the term refers. Some defi- nitions approach a general term by focusing on the attributes that determine the class. Each approach, as we shall see, has advantages and disadvantages. EXERCISES A.